The prevailing religion of the New England Colonies was Christianity, but along markedly different lines to the Anglican Church (or Church of England) that the settlers had fled. The Puritan approach to Christianity was to distance their practices from Catholicism, a task they considered Anglicanism to have failed at. Specifically, they rejected the episcopalian (hierarchical) structure of the Anglican Church, with its ranking of vicars, bishops and archbishops, and sought instead to implement a congregationalist structure whereby each church, or congregation, would be more or less independent and self-governing.
This congregationalist structure allowed churches to arrange their own taxes and to hire, as well as fire, their own ministers. They were also highly selective of their membership, despite the fact that church attendance was mandatory. The high standards of the Puritan Church made its survival through the generations difficult, and church leaders were forced to soften their requirements with the "Half-Way Covenant" in 1662. This allowed for baptized children who refused to give testimony to be granted partial membership, but not sainthood, which would have allowed them to vote on church affairs and take communion.
The religiously idealistic Puritans were also known as Non-Separatists in order to distinguish them from the more extreme, albeit much smaller, group of church reformists known as the Separatists. This group, perhaps more commonly known as the Pilgrims of the Plymouth Colony, had arrived before the Puritans, seeking a far more extreme break from Anglican Church traditions.